MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2013 Switzerland - Final Round
2013 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(7)
8
1
Hide problems
sum a^2 (a - b)(a + b) >0
Let
a
,
b
,
c
>
0
a, b, c > 0
a
,
b
,
c
>
0
be real numbers. Show the following inequality:
a
2
⋅
a
−
b
a
+
b
+
b
2
⋅
b
−
c
b
+
c
+
c
2
⋅
c
−
a
c
+
a
≥
0.
a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a} \ge 0 .
a
2
⋅
a
+
b
a
−
b
+
b
2
⋅
b
+
c
b
−
c
+
c
2
⋅
c
+
a
c
−
a
≥
0.
When does equality holds?
9
1
Hide problems
p^m - q^3 = n^3
Find all quadruples
(
p
,
q
,
m
,
n
)
(p, q, m, n)
(
p
,
q
,
m
,
n
)
of natural numbers such that
p
p
p
and
q
q
q
are prime and the the following equation is fulfilled:
p
m
−
q
3
=
n
3
p^m - q^3 = n^3
p
m
−
q
3
=
n
3
6
1
Hide problems
two non-empty stacks of n and m coins on a table
There are two non-empty stacks of
n
n
n
and
m
m
m
coins on a table. The following operations are allowed:
∙
\bullet
∙
The same number of coins are removed from both stacks.
∙
\bullet
∙
The number of coins in a stack is tripled. For which pairs
(
n
,
m
)
(n, m)
(
n
,
m
)
is it possible that after finitely many operations, no coins are more available?
1
1
Hide problems
{gcd(a,b),gcd(b,c),gcd(c,a), lcm(a,b), lcm(b,c), lcm(c, a)}= {2, 3, 5, 30, 60}
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of natural numbers such that the sets
{
g
c
d
(
a
,
b
)
,
g
c
d
(
b
,
c
)
,
g
c
d
(
c
,
a
)
,
l
c
m
(
a
,
b
)
,
l
c
m
(
b
,
c
)
,
l
c
m
(
c
,
a
)
}
\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}
{
g
c
d
(
a
,
b
)
,
g
c
d
(
b
,
c
)
,
g
c
d
(
c
,
a
)
,
l
c
m
(
a
,
b
)
,
l
c
m
(
b
,
c
)
,
l
c
m
(
c
,
a
)}
and
{
2
,
3
,
5
,
30
,
60
}
\{2, 3, 5, 30, 60\}
{
2
,
3
,
5
,
30
,
60
}
are the same.Remark: For example, the sets
{
1
,
2013
}
\{1, 2013\}
{
1
,
2013
}
and
{
1
,
1
,
2013
}
\{1, 1, 2013\}
{
1
,
1
,
2013
}
are equal.
4
1
Hide problems
f ( x / (y + 1)) = 1 - xf(x + y)
Find all functions
f
:
R
>
0
→
R
>
0
f : R_{>0} \to R_{>0}
f
:
R
>
0
→
R
>
0
with the following property:
f
(
x
y
+
1
)
=
1
−
x
f
(
x
+
y
)
f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)
f
(
y
+
1
x
)
=
1
−
x
f
(
x
+
y
)
for all
x
>
y
>
0
x > y > 0
x
>
y
>
0
.
2
1
Hide problems
p_1^2 + p_2^2 + · · + p_n^2 > n^3
Let
n
n
n
be a natural number and
p
1
,
.
.
.
,
p
n
p_1, ..., p_n
p
1
,
...
,
p
n
distinct prime numbers. Show that
p
1
2
+
p
2
2
+
.
.
.
+
p
n
2
>
n
3
p_1^2 + p_2^2 + ... + p_n^2 > n^3
p
1
2
+
p
2
2
+
...
+
p
n
2
>
n
3
5
1
Hide problems
2n + 1 students chooses a finite set of consecutive integers
Each of
2
n
+
1
2n + 1
2
n
+
1
students chooses a finite, nonempty set of consecutive integers . Two students are friends if they have chosen a common number. Everyone student is friends with at least
n
n
n
other students. Show that there is a student who is friends with everyone else.